Factor 12x² + 83x + 20

Here we will show you how to factor the quadratic function 12x² + 83x + 20 using the box method. In other words, we will show you how to factor 12x squared plus 83x plus 20 (12x^2 + 83x + 20) using the box method. It is a 5-step process:

Step 1: The standard form of a quadratic equation is ax² + bx + c. We start by labeling the different parts of our equation 12x² + 83x + 20, like this:

a = 12
b = 83
c = 20

Step 2: Next, we need to draw a box and divide it into four squares:

1	3x	20
4x	12x²	80x
	3x	20

We put 12x² (a) in the bottom left square and 20 (c) in the top right square, like this:

1	3x	20
4x	12x²	80x
	3x	20

Step 3: In order to fill in the other two squares, we need to do some math. We have to find two numbers that multiply together to give us the product of 12 times 20 (a × c), and add together to equal 83 (b).

More specifically, 12 times 20 is 240. Therefore, we need to find the two numbers that multiply to equal 240, and add to equal 83.

? × ? = 240
? + ? = 83

After looking at this problem, we can see that the two numbers that multiply together to equal 240, and add together to equal 83, are 3 and 80, as illustrated here:

3 × 80 = 240
3 + 80 = 83

Now, we can fill in the last two squares in our box with 3x and 80x. Place 3x in the upper left square, and place 80x in the lower right square.

1	3x	20
4x	12x²	80x
	3x	20

Step 4: Next, we need to find four final numbers to finish factoring our equation. We can see that our box is a 2 x 2 table made up of rows and columns.

Our goal is to find the numbers that, when multiplied together, result in the products in each square. We can do this by finding the greatest common factor in each row.

Let’s look at the top row. We have the terms 3x and 20. The greatest common factor of 3x and 20 is 1. Therefore, we write 1 to the left of the top row. You can see it here in the color green:

1	3x	20
4x	12x²	80x
	3x	20

Next, let’s look at the bottom row. We have the terms 12x² and 80x. The greatest common factor of 12x² and 80x is 4x. Therefore, we write 4x to the left of the bottom row. You can see it here in the color blue:

1	3x	20
4x	12x²	80x
	3x	20

To find the values below the table, we first divide 12x² by 4x (labeled in blue). This gives us 3x.

12x² ÷ 4x = 3x

You can see this value colored in orange below:

1	3x	20
4x	12x²	80x
	3x	20

Next, we divide 80x by 4x (labeled in blue). This gives us 20.

80x ÷ 4x = 20

You can see this value colored in purple below:

1	3x	20
4x	12x²	80x
	3x	20

Step 5: Now we have all of the information we need to finish factoring the equation. The values outside of the box are used to factor 12x² + 83x + 20. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get the answer:

(4x + 1)(3x + 20)

That’s it! Now you know how to factor the equation 12x² + 83x + 20.

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